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Colloquium on Topology, Gyula, Hungary
August 9-15, 1998
János Bolyai Mathematical Society
Budapest, Hungary |
|
Organizers
M. Bognár, A. Császár (chairman), J. Gerlits, I. Juhász, E. Makai, G. Moussong, R. Rimányi, L. Soukup, A. Stipsicz, J. Szenthe, A. Szücs (secretary)
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Compositions of simple maps
by
Jerzy Krzempek
University of Silesia, Katowice, Poland
Compositions of simple maps
Jerzy Krzempek (Katowice)
A map (= continuous function) is of order <= k if no its
point-inverse has more than k elements. Maps of order <= 2 are
called simple.
The following problem originates in papers of K. Borsuk and R. Molski
[Fund. Math. 45 (1957), 84-98] and K. Sielkucki [Fund. Math. 48
(1960), 217-228]: Which maps are compositions of simple closed
[respectively: open, clopen] maps? How many simple maps are really needed
to represent a given map?
-
[
-
] Theorem.
Every closed [open, clopen ] map f:X[( onto) || ( --> )]Y of order <= k
defined on a zero-dimensional metric space X is a composition
X=Xk[( fk-1) || ( --> )] ... [( f1) || ( --> )]X1=Y
of k-1 simple closed [open, clopen ] surjective maps
f1, ... , fk-1.
Moreover, these maps can be choosen so that every inverse
(f1 o ... o fi)-1(y) has exactly
min{i+1, |f-1(y)|} elements, y in Y and i=1, ... , k-1.
This upperbound of the number of simple maps is the least possible. The
foregoing theorem is a good way to Nagami's result concerning the
sharpness of the theorem on dimension-raising maps.
-
[
-
] Theorem.
Every closed map of order <= k with an n-dimensional metric domain
is the composition of (n+1)k-1 simple closed maps.
These theorems fail to be true on non-metrizable spaces. There exists
an appropriate map on a Cantor cube of uncountable weight.
Author's address : Institute of Mathematics, University of Silesia,
ul. Bankowa 14, PL-40-007 Katowice, POLAND
E-mail : krzempek@ux2.math.us.edu.pl
1991 Mathematics Subject Classification: 54E40, 54C10, 54F45.
Date received: July 27, 1998
Copyright © 1998 by the author(s).
The author(s) of this document and the organizers of the conference
have granted their consent to include this abstract in
Atlas Conferences Inc.
Document # cabc-48.